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Factors

Numbers · Factors

Syllabus tag: KCSE | Mathematics | Form 1 | Topic 2 Factors

Lesson objectives

By the end of this topic, you should be able to:

  • List the factors of a given number.
  • Express composite numbers as a product of prime factors.
  • Write prime factors in index (power) form.
  • Use factor trees and repeated division to find prime factors.

Factors

A factor of a number divides into it exactly, with no remainder.

The factors of 12 are 1, 2, 3, 4, 6 and 12.

a) Listing factors

Work in pairs, starting from 1.

For 24: 1 × 24, 2 × 12, 3 × 8, 4 × 6. Then 5 does not divide, and 6 has already appeared.

Stop when the pairs start to repeat. That happens once you pass the square root.

Listing in pairs means you never miss one.

b) Prime factors

A prime factor is a factor that is also a prime number.

Every composite number can be written as a product of primes, and in only one way. That is why prime factorisation is so useful.

c) The factor tree

Building a factor tree Building a factor tree Start with 60 divide by the smallest prime 60 = 2 × 30 30 is not prime, keep going 30 = 2 × 15 15 is not prime either 15 = 3 × 5 both are prime, so stop

Divide by the smallest prime that goes in. Write the two factors below.

Carry on with any factor that is not yet prime.

Stop when every branch ends in a prime.

The primes at the ends of the branches are the prime factors.

d) Repeated division

An alternative method. Divide by the smallest prime that goes in, write the quotient below, and repeat.

For 60: divide by 2 to get 30, then by 2 again to get 15. Then by 3 to get 5, and by 5 to get 1.

The divisors used are the prime factors: 2, 2, 3 and 5.

Both methods give the same answer. Use whichever you find clearer.

e) Index form

Writing in index form Writing in index form 60 = 2 × 2 × 3 × 5 all the prime factors the two 2s = 2² count the repeats 60 = 2² × 3 × 5 index form check → 4 × 3 × 5 = 60 multiply back

When a prime repeats, write it once with a small number showing how many times.

So 2 × 2 is written 2².

Index form is shorter and makes GCD and LCM work much easier later.

Always check by multiplying back. If you do not return to the original number, a factor was lost.

f) Where this is used

Simplifying fractions. Finding GCD and LCM. Simplifying surds later on. Any problem needing a number broken into its building blocks.

Words to know

  • Factor -- a whole number that divides another exactly, leaving no remainder.
  • Prime factor -- a factor that is itself a prime number.
  • Prime factorisation -- writing a number as a product of primes.
  • Index (power) -- the raised number showing how many times a base is multiplied by itself.
  • Composite number -- a number greater than 1 that is not prime.

:::checkpoint Check yourself

  1. List all the factors of 36.
  2. Express 84 as a product of prime factors.
  3. Write your answer in index form.
  4. Why does every composite number have only one prime factorisation? :::

Bridge to practice

The exercises begin with listing factors and identifying prime factors, move through the factor tree and repeated division methods, and finish with index form and counting factors. For any number above about 50, use repeated division rather than a tree — it is faster and far less prone to a missed branch.

Check yourselfPractise Factors — 10 questions →Next in MathematicsDivisibility Tests →